Specificity and robustness in transcription control networks Anirvan M. Sengupta*, Marko Djordjevic†, and Boris I. Shraiman*‡ *Bell Laboratories, Lucent Technologies, Murray Hill, NJ 07974; and †Department of Physics, Columbia University, New York, NY 10025 Edited by David R. Nelson, Harvard University, Cambridge, MA, and approved November 8, 2001 (received for review July 26, 2001) Recognition by transcription factors of the regulatory DNA ele- determine the binding probability. The interaction of a tran- ments upstream of genes is the fundamental step in controlling scription factor with a DNA sequence x (of length L) may be ϭ ⅐␧ ϵ͚L gene expression. How does the necessity to provide stability with approximated (9, 10) by the binding energy E(x) x iϭ1 4 ␤ ␤ ␤ respect to mutation constrain the organization of transcription ͚␤ϭ ␧ x where ␧ is the interaction energy with base ␤ at 1 i i ϭ i ␤ ϭ control networks? We examine the mutation load of a transcrip- position i 1,..., L of the DNA string and xi 1ifthe tion factor interacting with a set of n regulatory response elements sequence x contains base ␤ at position i and is 0 otherwise. Thus ␧␤ as a function of the factor͞DNA binding specificity and conclude on DNA binding properties of a factor are parametrized by i . theoretical grounds that the optimal specificity decreases with n. It is useful to consider the distribution of binding energies E(x) among all possible sequences, which is described by a histogram The predicted correlation between variability of binding sites (for ␳ a given transcription factor) and their number is supported by the (or density of states) (E) as shown in Fig. 2. The vast majority genomic data for Escherichia coli. The analysis of E. coli genomic of random sequences fall into the approximately Gaussian center data was carried out using an algorithm suggested by the bio- of this distribution, whereas the strongest binding, or consensus, sequence defines the leftmost edge (see Appendix A). (Note: It physical model of transcription factor͞DNA binding. Complete is convenient to set the scale of energy by the standard deviation results of the search for candidate transcription factor binding sites ͞͞ ͞ϳ ͞ of the binding energy in a random sequence ensemble and to set are available at http: www.physics.rockefeller.edu boris the average energy—i.e. the energy of nonspecific interac- ͞ ࿝ public search ecoli. tion—as E ϭ 0.) In equilibrium, the probability of any string x to bind a factor ͑ ͑ ͒ Ϫ ␮͒͞ he accumulation of knowledge on control of transcription in is given explicitly by f(E(x)) ϭ [e E x kBT ϩ 1]Ϫ1, where ␮ is Tsimple and complex organisms (1–4) poses many questions the chemical potential set by factor concentration. Provided that regarding its system-level function and organization. What as- the characteristic scale of binding energy is large compared to ͞ ϳ pects of control network architectures insure their stability with kBT [e.g., for lac-repressor (11) Em kBT 10] one may as a first respect to mutation along with their ability to adapt and acquire approximation replace f(E) by a step function; i.e. an ‘‘all or new function (see ref. 1)? In its turn, could better understanding nothing’’ binding condition. The binding condition identifies the of the general organization of these networks help to dissect subset of all possible sequences (of length L) that at a given specific systems? Motivated by these questions, we formulate a physiological concentration of a factor bind to it strongly: sequence, x, belongs to such a subset for factor Fi if it satisfies model of transcription control that captures many of the essen- Ͻ ␮ tial features of the process and can be tested against data. This Ei(x) i. On Fig. 3, sequence subsets binding to different model is applied to the study of evolutionary stability of bacterial factors are pictured as nonoverlapping discs. Points within a given disk denote response elements controlled by F : their regulons (5), each involving a transcription factor that controls i number, ni, is the degree of pleiotropy of the factor. A given multiple genes by binding to multiple regulatory sequence factor, present in certain concentration, may be characterized by elements. Evolutionary stability, or stability with respect to its ‘‘binding specificity,’’ ␴ ϵ ln vϪ1, defined in terms of the mutation, provides a sensible quantitative definition of robust- fraction, v,ofrandom sequences (of length L) that bind strongly. ness (1, 6, 7). Factors with highly sequence-specific binding These sequences lie in the ‘‘tail’’ of energy distribution ␳(E) impose severe constraint on regulatory sequences, increasing the below ␮ (see Fig. 2). In Fig. 3 binding specificity is represented probability of failure due to mutation, whereas low specificity of by disk area. binding increases the probability of spurious interactions. Ro- In addition to regulatory response elements, DNA regions bustness is maximized by the compromise between these two upstream of genes contain stretches of sequence without direct effects. From this follows a quantitative prediction relating the function. Some of these nonfunctional stretches are truly passive; number of elements in a regulon and the degree of binding site others, however, may become ‘‘active’’ if through mutation they sequence variability. Our model also suggests a method of acquire a binding site for some transcription factor that can then identifying candidate transcription factor binding sites, which we interfere, positively or negatively, with transcription. In Fig. 3, use in the analysis of the genomic data for Escherichia coli (8). such ‘‘potential cis-regulatory sites’’ are represented as points Genomic data provides support for the prediction of the theory. outside the disks. Model of Transcription Factor͞DNA Interaction Mutation Load in Transcription Control In modeling the transcription control network, let us concentrate Let us now introduce mutations. An accumulation of point on the flow of information from factors to genes. Active mutations in a response element sequence corresponds to a transcription factors bind to the regulatory response element (RE) subsequences associated with genes (2). A given gene may This paper was submitted directly (Track II) to the PNAS office. be controlled by multiple repressing or activating factors acting through multiple REs (see Fig. 1). It will suffice for now to Abbreviation: RE, response element. ‡ assume that all of the controlling factors and REs are nonre- To whom reprint requests should be addressed at: 1-D236 Bell Labs, 700 Mountain Avenue, Murray Hill, NJ 07974. E-mail: [email protected]. dundant so that the loss of factor͞RE recognition results in a §The case of more complex and redundant regulatory interactions can be considered within significant detriment of fitness, due to the loss of regulatory the same framework. § linkage. The publication costs of this article were defrayed in part by page charge payment. This Binding of a factor to an RE depends on the factor concen- article must therefore be hereby marked “advertisement” in accordance with 18 U.S.C. tration and the binding energy of the pair which together §1734 solely to indicate this fact. 2072–2077 ͉ PNAS ͉ February 19, 2002 ͉ vol. 99 ͉ no. 4 www.pnas.org͞cgi͞doi͞10.1073͞pnas.022388499 Fig. 1. Schematic model of transcription control. Fs are active transcription factor proteins, xs are response element subsequences upstream of the coding regions of the genes, G. Arrows indicate regulatory interactions. random walk of the RE in sequence space (Fig. 3). A mutation in the DNA binding domain of the transcription factor changes its interaction energy with the sequence, ␧. This change would appear in Fig. 3 as a random shift of the disc representing the binding subset in the sequence space. Either process could lead to an RE moving out of the binding subset of sequences, disrupting the interaction with the factor. In our model of a nonredundant network involving only essential genes, such failure is assumed to be ‘‘lethal.’’ Another form of regulatory failure involves a mutation of the potential regulatory site, causing it to enter the domain of binding with a wrong factor (see Fig. 3. Response elements (dots) and factor binding subsets (disks) in Fig. 3). For simplicity, this process is assumed to be lethal as well. sequence space. RE located within a disk binds the corresponding factor. Black EVOLUTION To discuss the robustness of any given network, we must be able dots lying outside the discs represent potential cis-regulatory sites, which must not bind transcription factors in order to avoid interference with transcription to calculate the probability (per mutation) of both modes of control. Arrows represent random changes in the sequence of REs (and of failure. potential cis-regulatory sites) due to mutation. For a single factor͞RE link, the binding probability depends only on the interaction energy E(x) ϭ ␧⅐x which changes randomly with mutations in x and ␧. Although this is a discrete which governs the time evolution of n(E, t). It has the form of process, most qualitative features could be understood in the biased diffusion in the energy variable: limit where the energy changes continuously. This approxima- Ѩ ͑ ͒ ϭѨ2 ͑ ͒ ϩѨ ͓ ͑ ͔͒ tion is accurate when the binding energy has contribution from tn E, t En E, t E En E, t [1] many sites, i.e. L is large, and mutation at an individual site results in only a small change in the total energy. Consider a where unit of time is set by the point mutation rate. In addition ͉ ϭ mutating ‘population’ of single factor͞RE links with n(E, t) we impose the boundary condition n(E, t) Eϭ␮ 0 which denoting the number of links with binding energy in the E, E ϩ implements the assumption that ‘‘escape’’ of the RE from the dE interval.